But The News understands that Rotherham, Wycombe and Hull City are also interested in the club vice-captain.
Brown's contract expired at the end of the Blues' League One play-off defeat to Oxford United earlier this month.
The left-back has since been locked in negotiations with the club over extending his stay on the south coast.
It's understood that talks between both parties have been progressing well and the deal is around 70 per cent down the line.
However, Brown isn't short of suitors and does have the opportunity to move to the Championship.
Newly-promoted Wycombe and Rotherham have both declared their interest in the Farnborough-born full-back.
The Chairboys' skipper Joe Jacobson is currently first choice at Adams Park but turns 34 in November.
Meanwhile, Rotherham boss Paul Warne has come up against Brown on several occasions and knows his attributes well.
Relegated Hull are also admirers. Grant McCann is keen to add the ex-QPR man to his squad as part of the Tigers' League One promotion push.
Arriving on a free transfer from Bristol Rovers in June 2018, Brown has featured 72 times for Kenny Jackett's side, scoring one goal.
He helped Pompey clinch the Checkatrade Trophy at Wembley during the 2018-19 season, as well as featuring in all but two league games.
Last term, Brown again started as first-choice left-back and was appointed vice-skipper to Tom Naylor in September, after Brett Pitman and Gareth Evans were stripped of their duties.
However, the 29-year-old was troubled by an ongoing Achilles problem during the first half of the campaign, which forced him to undergo surgery shortly before Christmas.
That meant Pompey brought Steve Seddon in on loan from Birmingham in January and Brown was unable to win back a starting spot when he did regain his fitness.
However, the former Salisbury loanee featured in both legs of the play-off defeat to Oxford and was arguably the Blues' best player across the two games.
Jackett’s side currently have no senior left-back under contract.
Seddon has returned to Birmingham, while Brandon Haunstrup turned down a new contract to join Scottish Premiership side Kilmarnock.
Joe Hancott will return to the club to complete his third-year scholarship.